Nit of the Day: Drawing Dead and Raising the River

[DavidC]

Let me introduce to you, Nit of the Day, which definitely won't be produced daily, but hopefully will be made no more than once per day. I'll be talking about errors that I see in posters' thinking, which will sometimes have an extremely small -EV impact on their game, due to the rarety of when they'll be wrong about generalizations that they've made. At the same time, seeing things like this drive me nuts, so I'm posting about them. [img]/images/graemlins/smile.gif[/img]

If I'm wrong in any of these, or if you have something to contribute which adds to our understanding of the issue, that'd be awesome!

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A few days ago, a new poster posted a hand where they called on the turn on a paired board, trying to hit a flush, and then they called a river bet instead of raising the river when they hit. They did this because they thought that they were drawing dead. Someone told them that if you're not going to raise the river, you should fold your four-flushes on the turn.

This isn't always correct. On the river, you evaluate calling vs raising based on a number of things. On the turn, you evaluate calling with a drawing hand based a number of things. In the case of calling on the turn when you think you may be drawing dead even if you improve, it's possible for a really large pot to give you good enough odds to call when you think you're drawing dead, while on the river you wouldn't have enough value to raise, due to reverse implied odds.

For the purposes of example, I'll say a 1,000,000 bb pot on the turn, with a 99% chance of drawing dead and a 25% chance of hitting your flush. It's worth it to call on the turn, it's worth it to call on the river, and it's not worth it to raise on the river.

I'm kinda curious about how large the pots have to be, and how low the chance of drawing dead has to be, before you could actually make a case for calling on the turn and the river. The reasoning behind calling on the river is the same reasoning behind calling on the turn, though: you're getting good pot-odds for your money. The reason that you don't raise the river is the same reason that you don't raise the turn: you're losing more money with each further bet that goes into the pot.

When you do hit your river flush, does your expectation for winning the hand diminish when your opponent shows no regard and bets into you anyway?

[bozlax]

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When you do hit your river flush, does your expectation for winning the hand diminish when your opponent shows no regard and bets into you anyway?

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Doesn't that depend on, uh, hmmm, what's that word, again? Oh, yeah, "reads."

[bozlax]

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I'm kinda curious about how large the pots have to be, and how low the chance of drawing dead has to be, before you could actually make a case for calling on the turn and the river.

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I'm not sure you stated this correctly. Don't you mean, how high the chance you're drawing dead is? Because, if the chance you're drawing dead is low then you're going to be raising the river.

So, with a four-flush on the turn, you need 5BB in the pot to call a single bet, right? The pot on the river will therefore be 7BB when it's bet into you. If you think there's greater than a 14% chance you're drawing dead, you can't call, or, to state it more generally, if you think there's a greater than 1/pot% chance you're drawing dead, you can't call. Obviously, the larger the pot, the lower % of the time you have to be good to call.

If, otoh, you think there's greater than a 50% chance you're not dead, you raise.

So, anything between 1/pot% and 51% is a call on the river.

Beyond that, it's read-dependent.

Yeah?

[DavidC]

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I'm not sure you stated this correctly. Don't you mean, how high the chance you're drawing dead is?

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We should focus on the numbers here, but I like your terminology best.

--Dave.

[DavidC]

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The pot on the river will therefore be 7BB when it's bet into you. If you think there's greater than a 14% chance you're drawing dead, you can't call, or, to state it more generally, if you think there's a greater than 1/pot% chance you're drawing dead, you can't call.

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On the river, if it's a 6bb river, he bets and it's 7bb, and then you're thinking about calling, your break-even point is when you win 1/8 of the time, not when you win 1/7 of the time (if you win 1/7, you win money by calling).

Consider: River aggressor bets 1/4 pot, you therefore have to win 1/5 of the time. (25% of pot called requires 20% success). 25/125 = 1/5. Therefore:

success requirement = call% / (100 + call%)

[bozlax]

[ QUOTE ]
On the river, if it's a 6bb river, he bets and it's 7bb, and then you're thinking about calling, your break-even point is when you win 1/8 of the time, not when you win 1/7 of the time (if you win 1/7, you win money by calling).

Consider: River aggressor bets 1/4 pot, you therefore have to win 1/5 of the time. (25% of pot called requires 20% success). 25/125 = 1/5. Therefore:

success requirement = call% / (100 + call%)

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You nit. You're right, of course, and then you go and eff up your example numbers. If Villan bets 1/4 pot, then you're only calling 20% of the pot, so it's 1/6. Then you redeem yourself with your general statement.

[DavidC]

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If, otoh, you think there's greater than a 50% chance you're not dead, you raise.

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This is something that I'm interested in learning, because it's common sense that you're right about this, but you're wrong. I'm pretty sure that you know this, but I'll go over it just so we don't miss anything when putting together our formula. Actually, you'd be pretty damn close to correct if hero had 2bb or less remaining, or if villain only had 1bb left in his stack.

The formula for raising the river heads-up is easier than the one for determining if you should bet the river out of position or not. Could you be so kind as to make the formula for that one, since I did this one? I'd be happy to nit the hell out of it when you're done, so that we can get as close to a true answer as possible...

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Let:
A) chance that you're ahead and he calls
B) chance that you're ahead and he raises and you call
C) chance that you're behind and he raises and you call
D) chance that he's ahead and he calls your raise
E) chance that you're ahead and he folds
F) chance that you're behind and he folds
G) chance that you both have the same hand and he folds.

For the purposes of this discussion, I'm going to say that F and G never happen. If it does, though, you have to understand that the factor by which you multiply F is the pot size, and G is half the pot size, whereas other factors are multiplied as shown below. Therefore in large pots, when F and G exist, they have a very large impact on the EV even at low values of F and G. Since E has factor of zero, it's removed from the formula, but it's taken into consideration when creating the values of A through D.

Also, even though the NLHE equivalent of this formula is The One True Formula (which can be applied to limit), it would make my brain hurt, so we're assuming both opponents are playing with infinite stacks in a LHE game.

EV = 1A+2B-2C-1D*
* please note that I'm using 2B and 2C because of the instant profit/loss of 1bb occuring no matter what

Trial with your C value shows:
Let: A=48%, B=1%, C=49%, D=1%, E=1%
= 1(0.48) + 2(0.01) - 2(0.49) - 1(0.01)
= 0.48 + 0.02 - 0.98 - 0.01
= -0.49BB as a result of raising the river when C = 50%.

Therefore if there's a 50% chance that your opponent is ahead, you should definitely not be betting, according to this formula.

Edit: in this scenario, if villain three-bets us, we should fold unless the pot is greater than 99 bets, so we may need to tweak the formula under the assumption that we fold if he three-bets. Or we need to tweak the values to make a high enough value for him three-betting with a beaten hand that it makes sense to call him on the river, or we have to accept that we're just not good enough to fold the river when three-bet, even though we know we're not getting good enough odds to call... regardless, we have to tweak the numbers a bit to determine where the break-even point is, in order to achieve expert river play.

[DavidC]

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You nit. You're right, of course, and then you go and eff up your example numbers... Then you redeem yourself with your general statement.

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I [img]/images/graemlins/heart.gif[/img] Boz. [img]/images/graemlins/cool.gif[/img]

You give me a harder time here than anyone, but I appreciate it. [img]/images/graemlins/smile.gif[/img]

FWIW, I made a longer reply, cut it down for readability, and lost the section where I acknowledged that you knew this already. However, the actual value of the number IS important, if we're to answer the question of whether you should be calling the turn at all.

[DavidC]

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When you do hit your river flush, does your expectation for winning the hand diminish when your opponent shows no regard and bets into you anyway?

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This is reads based. If we're playing a great opponent, who will be betting any reasonable hand into us on the river when he's out of position, then it doesn't change. If we're playing a lag, it doesn't change, and if we're playing a fishy type, it may change, depending on reads.

You've just opened up a huge can of worms here. I suppose it's possible, vs certain opponents, that we should call the turn, hit our flush, and fold to a river bet. That would have to be taken into the EV calculations on the turn, which would affect our decision on whether or not to call the turn at all.

I think Shill spoke about this in a post about calling with with TPTK on the turn vs turn checkraises, saying that it's -EV to call the turn if you will also call the river, but that it's +EV to call the turn and then call the river only if you improve. IIRC, raising the river was not discussed in that post, but I don't believe that the possibility of an opponent bluffing was.

Edit: Although I haven't crunched the numbers on this, I think it actually makes it easier to call the turn if we can fold to a river bet, because our EV will necessarily be higher on the river, because we'll never have to call when we're losing.